Theorem pnrmtop 23459

Description: A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
pnrmtop	⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)

Proof of Theorem pnrmtop

Step	Hyp	Ref	Expression
1		pnrmnrm 23458	. 2 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)
2		nrmtop 23454	. 2 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
3	1, 2	syl 18	1 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2145  Topctop 23011  Nrmcnrm 23428  PNrmcpnrm 23430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-cnv 5660  df-dm 5662  df-rn 5663  df-iota 6481  df-fv 6533  df-ov 7403  df-nrm 23435  df-pnrm 23437

This theorem is referenced by:  pnrmopn  23461